Re: The Correct Way to Hold a Knife
Different styles do different things. Like Mark said, if you really want to get into it, the Filipino arts have extensive techniques.
All that said, in Japanese arts the "correct" way to hold a knife to attack still depends completely on context of the overall style of "tantojutsu" being employed (to keep it focused in Japanese arts). In forays into koryu stuff I've learned mostly the so-called "hammer grip" (or Saber Grip if you're letting the thumb float more) and the so-called reverse edge out grip.
Another grip you'll sometimes see is the so-called "icepick" or what we only somewhat jokingly call the "crazed, jealous wife" grip. That's a reverse grip, edge in. Think "Psycho" with the knife stabbing down repeatedly.
FWIW some of the "best" knifework I've seen has been those who use the "standard" grip with finesse (Toby Threadgil does some amazing stuff at an incredible speed from this grip) or in the reverse edge out (see Toby again for this stuff).
Anyway, the point is that "correct" is generally contingent on a larger framework. Many arts have "mirrored" empty hand and tanto arts where the same overall movements are used in order to have an efficient means of skills transmission. As such that can affect the answer as to which grip is correct even within different techniques.
Bottom line is that used well any grip can be deadly and very difficult at best to deal with. So it is good to train in a variety of attacks.
As an aside, years ago I went to a seminar in a style of Aikido not my own. The instructor called me up with a tanto. I reflexively grabbed it into reverse edge out grip in my right hand and stepped up, left foot forward. He smiled and said "Ah, you've done knife work before." He told the seminar that if you see someone carrying a knife like that and they move toward you, run... He then had me change to a right hand hammer grip for a standard "Aikido" tsuki.